The figure below shows four small charged spheres arranged at the corners of a square with side

d = 25.0 cm on each side.(Let q1 = +3.00 nC, q2 = +9.00 nC, q3 = +6.00 nC, and q4 = +5.00 nC. Assume q3 is located at the origin and +x axis is to the right and the +y axis is up along the page. Express your answers in vector form.)
(a) What is the electric field at the location of the sphere with charge +6.00 nC? E =--- N/C
(b) What is the total electric force exerted on the sphere with charge +6.00 nC by the other three spheres?F =---N

Not certain where the other charges are, it matters, as you are adding vectors.

To find the electric field at the location of the sphere with charge +6.00 nC, we need to calculate the individual electric fields caused by each of the other three spheres and then add them up.

(a) Electric field at the location of the sphere with charge +6.00 nC:

Step 1: Calculate the electric field due to q1.
The electric field due to a point charge can be found using the formula:
E = k * q / r^2
where E is the electric field, k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge.

The distance between q1 and the sphere with charge +6.00 nC is the diagonal of the square, which can be found using the Pythagorean theorem:
d = sqrt(25^2 + 25^2) = 25 * sqrt(2) cm

Now, we can calculate the electric field due to q1:
E1 = k * q1 / r^2
E1 = (8.99 x 10^9 Nm^2/C^2) * (3.00 x 10^-9 C) / (25 * sqrt(2) cm)^2

Step 2: Calculate the electric field due to q2.
The distance between q2 and the sphere with charge +6.00 nC is also the diagonal of the square, which is the same as before:
d = 25 * sqrt(2) cm

Now, we can calculate the electric field due to q2:
E2 = k * q2 / r^2
E2 = (8.99 x 10^9 Nm^2/C^2) * (9.00 x 10^-9 C) / (25 * sqrt(2) cm)^2

Step 3: Calculate the electric field due to q4.
The distance between q4 and the sphere with charge +6.00 nC is the side length of the square, which is 25 cm.

Now, we can calculate the electric field due to q4:
E4 = k * q4 / r^2
E4 = (8.99 x 10^9 Nm^2/C^2) * (5.00 x 10^-9 C) / (25 cm)^2

Step 4: Add up the electric fields.
The total electric field at the location of the sphere with charge +6.00 nC is the vector sum of the electric fields due to q1, q2, and q4.
E_total = E1 + E2 + E4

Now you can substitute the values and calculate E_total.

(b) To find the total electric force exerted on the sphere with charge +6.00 nC by the other three spheres, we can use the formula for the electric force between two point charges:
F = k * |q1| * |q2| / r^2
where F is the electric force, k is the Coulomb's constant, q1 and q2 are the charges, and r is the distance between them.

Step 1: Calculate the force due to q1.
The force due to q1 can be found using the above formula:
F1 = k * |q1| * |q3| / r^2
F1 = (8.99 x 10^9 Nm^2/C^2) * (3.00 x 10^-9 C) * (6.00 x 10^-9 C) / (25 * sqrt(2) cm)^2

Step 2: Calculate the force due to q2.
The force due to q2 can be found using the same formula:
F2 = k * |q2| * |q3| / r^2
F2 = (8.99 x 10^9 Nm^2/C^2) * (9.00 x 10^-9 C) * (6.00 x 10^-9 C) / (25 * sqrt(2) cm)^2

Step 3: Calculate the force due to q4.
The force due to q4 can be found using the same formula:
F4 = k * |q4| * |q3| / r^2
F4 = (8.99 x 10^9 Nm^2/C^2) * (5.00 x 10^-9 C) * (6.00 x 10^-9 C) / (25 cm)^2

Step 4: Add up the forces.
The total electric force exerted on the sphere with charge +6.00 nC by the other three spheres is the vector sum of the forces due to q1, q2, and q4.
F_total = F1 + F2 + F4

Now you can substitute the values and calculate F_total.